Solve for $x$ : $ 3|x - 5| - 3 = 4|x - 5| + 7 $
Subtract $ {3|x - 5|} $ from both sides: $ \begin{eqnarray} 3|x - 5| - 3 &=& 4|x - 5| + 7 \\ \\ {- 3|x - 5|} && {- 3|x - 5|} \\ \\ -3 &=& 1|x - 5| + 7 \end{eqnarray} $ Subtract $7$ from both sides: $ \begin{eqnarray} -3 &=& 1|x - 5| + 7 \\ \\ {- 7} && {- 7} \\ \\ -10 &=& 1|x - 5| \end{eqnarray} $ Simplify: $ -10 = |x - 5| $ The absolute value cannot be negative. Therefore, there is no solution.